A proximal point algorithm for DC functions on Hadamard manifolds

Transcription

1 Noname manuscript No. (will be inserted by the editor) A proimal point algorithm for DC functions on Hadamard manifolds J.C.O. Souza P.R. Oliveira Received: date / Accepted: date Abstract An etension of a proimal point algorithm for difference of two conve functions is presented in the contet of Riemannian manifolds of nonposite sectional curvature. If the sequence generated by our algorithm is bounded it is proved that every cluster point is a critical point of the function (not necessarily conve) under consideration, even if minimizations are performed ineactly at each iteration. Application in maimization problems with constraints, within the framework of Hadamard manifolds is presented. Keywords Nonconve optimization proimal point algorithm DC functions Hadamard manifolds Mathematics Subject Classification (2000) 49M30 90C26 90C48 1 Introduction It is well known that the class of Proimal Point Algorithm (PPA) is one of the most studied methods for finding zeros of maimal monotone operators and, in particular it s used to solve conve optimization problems. The classical PPA was introduced into optimization literature by Martinet [1]. It is based on the notion of proimal mapping J f λ, J f λ () = arg min z R n{f(z) + 1 2λ z 2 }, (1) This research was partially supported by CNPq, Brazil. J.C.O. Souza COPPE-Sistemas, Universidade Federal do Rio de Janeiro, Caia Postal 68511, CEP , Rio de Janeiro, RJ, Brazil and CEAD, Universidade Federal do Piauí, Teresina, PI, Brazil joaocos.mat@ufpi.edu.br P.R. Oliveira COPPE-Sistemas, Universidade Federal do Rio de Janeiro, Caia Postal 68511, CEP , Rio de Janeiro, RJ, Brazil poliveir@cos.ufrj.br

2 2 J.C.O. Souza, P.R. Oliveira introduced earlier by Moreau [2]. The PPA was popularized by Rockafellar [3], who showed the algorithm converges even if the auiliary minimizations in (1) are performed ineactly, which is an important consideration in practice. The algorithm is useful, however, only for conve problems, because the idea underlying the results is based on the monotonicity of subdifferential operators of conve functions. Therefore, PPA for nonconve functions has been investigated by many authors (cf. [4],[5] and references therein). In Rockafellar [3], the algorithm starting with any 0 R n, iteratively updates k+1 conforming to the following recursion 0 T ( k+1 ) + k+1 k, (2) where 0 < c, is a sequence of scalars and T is a multivalued maimal monotone operator from R n to itself. On the other hand, etension to Riemannian manifolds of the concepts and techniques that fit in Euclidean spaces is natural and nontrivial. Actually, in recent years, some algorithms defined to solve minimization problems have been etended from Hilbert space framework to the more general setting of Riemannian manifolds (see, for eample [6]-[20]). The main advantages of these etensions are that nonconve problems in the classic sense may become conve and constrained optimization problems may be seen as unconstrained ones through the introduction of an appropriate Riemannian metric (see [6]-[10]). Numerical solution of optimization problems defined on Riemannian manifolds arise in a variety of applications, e.g., in computer vision, signal processing, motion and structure estimation, or numerical linear algebra (see for instance [21]-[24]). Also, these etensions give rise to interesting theoretical questions. To etend (1) and (2) to the contet of Riemannian manifolds was the subject of [6] and [11], respectively. We will consider a special class of nonconve optimization problem of the form min f() = g() h(), (3) M where g, h : M R are proper, conve and lower semi-continuous (lsc) functions and M is a complete Riemannian manifold. The function f is called a DC function (i.e. difference of two conve functions). The interest in the theory of DC functions has much increased in the last years (see for instance [25]-[29] and references therein), but only a few have proposed some specific algorithms or numerical eperiments (for eample [30]-[31]). Some mathematical reasons used to eplain interest in DC functions can be found in [26], for instance, the class of DC functions defined on a compact conve set X R n is dense in the set of continuous function over X, endowed with the topology of uniform convergence over X. Sun et al [30] proposed a proimal point algorithm for minimization of DC functions which use conve properties of the two conve functions separately. The purpose of this paper is to etend the PPA presented in [30] to Riemannian manifolds framework. Also, two different ineact methods of our algorithm are considered. Moreover, an application to the constrained optimization problems on Hadamard manifolds is given. To the best of our knowledge a proimal point algorithm to solve DC optimization problems in the contet of Riemannian manifolds has not been established yet. The paper is organized as follows. In Sect. 2, some fundamental definitions, properties and notations of Riemannian manifolds are presented. In Sect. 3, some definitions, notations and properties of conve analysis on Riemannian manifolds

3 A proimal point algorithm for DC functions on Hadamard manifolds 3 are presented. Convergence analysis of the eact version and ineact versions of the algorithm are provided in Sect. 4 and 5, respectively. In Sect. 6, an application to constrained optimization problems on Hadamard manifolds is given. 2 Basic Concepts In this section, we introduce some fundamental properties and notations of Riemannian manifold. These basics facts can be found in any introductory book of Riemannian geometry, for eample [32], [33]. Let M be a connected m-dimensional C manifold and let T M = {(, v) : M, v T M} be its tangent bundle, where T M is the tangent space of M at. T M is a linear space and has same dimension of M, moreover, because we restrict ourselves to real manifolds, it is isomorphic to R m. If M is endowed with a Riemannian metric g, then M is a Riemannian manifold and we denoted it by (M, g). The inner product of two vectors u and v in T M is written u, v := g (u, v) where g is the metric at the point. The norm of a vector u T M is defined by u := u, u 1/2. Recall that the metric can be used to define the length of piecewise smooth curve c : [a, b] M joining to, i.e., such that c(a) = and c(b) =, by L(c) = b a c (t) dt. Minimizing this length functional over the set of all such curves we obtain a Riemannian distance d(, ) which induces the original topology on M. Let be the Levi-Civita connection associated to (M, g). A vector field V along c is said to be parallel if c V = 0. If c itself is parallel we say that c is a geodesic. The geodesic equation γ γ = 0 is a second order nonlinear ordinary differential equation, then γ = γ v (, ) is determined by its position and velocity v at. It is easy to check that γ is constant. We say that γ is normalized if γ = 1. The restriction of a geodesic to a closed bounded interval is called a geodesic segment. A geodesic segment joining to in M is said to be minimal if its length equals d(, ) and this geodesic is called a minimizing geodesic. A Riemannian manifold is complete if geodesics are defined for any values of t. Hopf-Rinow s theorem asserts that if this is the case then any pair of points, say and, in M can be joined by a (not necessarily unique) minimal geodesic segment. Moreover, (M, d) is a complete metric space and bounded and closed subsets are compact. In this paper, all manifolds are assumed to be complete. Take M, the eponential map ep : T M M is defined by ep (v) = γ v (1, ). We denote by R the curvature tensor defined by R(X, Y ) = X Y Z Y X Z [Y,X] Z, where X, Y and Z are vector fields of M and [X, Y ] = Y X XY. Then the sectional curvature with respect to X and Y is given by K(X, Y ) = ( R(X, Y )Y, X )/( X 2 Y 2 X, Y 2 ), where X 2 = X, X. If K(X, Y ) 0 for all X and Y, then M is called a Riemannian manifold of nonpositive curvature and we use the short notation K 0. A complete simply connected Riemannian manifold of nonpositive sectional curvature is called a Hadamard manifold. The following result is well known (see, for eample [33], Theorem 4.1, p.221). Theorem 1 Let M be a Hadamard manifold and let p M. Then ep p : T p M M is a diffeomorphism, and for any two points p, q M there eists a unique normalized geodesic joining p to q, which is, in fact, a minimal geodesic.

4 4 J.C.O. Souza, P.R. Oliveira This theorem shows that M is diffeomorphic to the Euclidean space R n. Thus, we see that M has the same topology and differential structure as R n. Moreover, Hadamard manifolds and Euclidean spaces have some similar geometrical properties. One of the most important properties is described in the following theorem, which is taken from ([33], Proposition 4.5, p.223) and will be useful in our study. Recall that a geodesic triangle (p 1, p 2, p 3 ) of a Riemannian manifold is the set consisting of three distinct points p 1, p 2 and p 3 called the vertices and three minimizing geodesic segments γ i+1 joining p i+1 to p i+2 called the sides, where i = 1, 2, 3(mod3). Theorem 2 (Comparison theorem for triangles) Let M be a Hadamard manifold and ( 1, 2, 3 ) a geodesic triangle. Denote by γ i+1 : [0, l i+1 ] M geodesic segments joining i+1 to i+2 and set l i+1 := L(γ i+1 ), θ i+1 = (γ i+1(0), γ i(l i )), where i = 1, 2, 3(mod3). Then θ 1 + θ 2 + θ 3 π (4) l 2 i+1 + l 2 i+2 2l i+1 l i+2 cosθ i+2 l 2 i. (5) Let γ : [a, b] M be a normalized geodesic segment. A differentiable variation of γ is by definition a differentiable mapping α : [a, b] ( ɛ, ɛ) M satisfying α(t, 0) = γ(t). The vector field along γ defined by V (t) = ( α/ s)(t, 0) is called the variational vector field of α. The first variational formula of arc length on α is given as follows: L (γ) := d ds L(c s) s=0 = V, γ b a, (6) where c s (t) = α(t, s) with s ( ɛ, ɛ). The Riemannian distance plays a fundamental role in the net sections. We proceed now stating a result which we will go to use. Let M be a Hadamard manifold. For any M we can define the eponential inverse map ep 1 : M T M which is C. Since d(, ) = ep 1 (), then the map ρ : M R defined by ρ () = 1 2 d2 (, ) is C and its gradient at is gradρ () = ep 1 ( ) (see, [33]). Using the properties of the parallel transport and the eponential map, we obtain the following proposition that will be used in the net sections. Proposition 1 Let M be a Hadamard manifold. Let 0 M and { k } M be such that k 0. Then the following assertions hold. 1. For any y M, we have ep 1 k y ep 1 0 y and ep 1 y k ep 1 y If v k T km and v k v 0, then v 0 T 0M. 3. Given u k, v k T km and u 0, v 0 T 0M, if u k u 0 and v k v 0, then u k, v k u 0, v For any u T 0M, the function F : M T M defined by F () = P, 0u for each M is continuous on M. Proof See [11], Lemma 2.4, p. 666.

5 A proimal point algorithm for DC functions on Hadamard manifolds 5 3 Conveity on Riemannian Manifolds In this section, we introduce some definitions and notation of conveity on Riemannian manifolds. We also present some properties of the subdifferential of a conve function; see [34] for more details. A subset C M is said to be conve if, for any points p and q in C, the geodesic joining p to q is contained in C, that is, if γ : [a, b] M is a geodesic such that γ(a) = p and γ(b) = q, then γ((1 t)a + tb) C for all t [0, 1]. Let f : M R be a proper etended real-valued function. The domain of the function f is denoted by dom(f) and defined by dom(f) = { M : f() + }. The function f is said to be conve (respectively, strictly conve) if, for any geodesic segment γ : [a, b] M, the composition f γ : [a, b] R is conve (respectively, strictly conve), that is, (f γ)(ta + (1 t)b) t(f γ)(a) + (1 t)(f γ)(b), for any a, b R and 0 t 1. The subdifferential of f at is defined by f() = {u T M ; u, ep 1 y f(y) f(), y M}. (7) Then f() is a closed conve (possible empty) set. The proofs of the above assertions and the following propositions can be found in [6] and [34]. Proposition 2 Let { k } M a bounded sequence. If the sequence {v k } is such that v k f( k ) for each k N, then {v k } is also bounded. Proposition 3 Let M be a Hadamard manifold and let f : M R be conve function. Then, for any M, there is s T M such that f(y) f() + s, ep 1 y, y M. In other words, the subdifferential f() of f at M is nonempty. Proposition 4 If a function f : M R is conve, then for any M and λ > 0, there eists a unique point, denoted by p λ (), such that f(p λ ()) + λ 2 d2 (p λ (), ) = f λ () characterized by λ(ep 1 p λ () ) f(p λ()), where f λ () = inf y M {f(y)+λd 2 (, y)}. 4 Proimal Point Algorithm Let M be a Hadamard manifold and let f : M R be a DC function, i.e., f() = g() h(), where g, h : M R are proper, conve and lsc functions satisfying dom(g) dom(h). A necessary condition for to be a local minimum of f is that 0 f() g() h(). In other words, the subdifferentials g() and h() must overlap: g() h(). A similar condition holds true when is a local maimum of f. So, we will focus our attention on finding critical points of f. The set of critical points of f is defined by

6 6 J.C.O. Souza, P.R. Oliveira S = { M; h() g() }. Observe that a necessary and sufficient condition for a point to be a critical point of a DC function f is that 1 c ep 1 y g(), where y = ep (cw), for any w h() and c > 0 a real number. Throughout the remainder of this paper, we always assume that M is a Hadamard manifold, f : M R is a bounded from below DC function, such that f() = g() h(), and S. For finding critical points of a DC functions on Hadamard manifolds, which satisfies necessary optimality conditions, we consider the following algorithm: Algorithm (DCPPA) Step 1: Given an initial point 0 M and a bounded sequence of positive numbers { } [b, c]. Step 2: Compute w k h( k ) and set y k := ep k( w k ). (8) Step 3: Compute k+1 := arg min M {g() d 2 (, y k )}. (9) If k+1 = k, stop. Otherwise, k := k + 1 and return to Step 2. The well definition of the sequences { k } and {y k } follows immediately from Proposition 3 and 4. Note that when h() = 0, algorithm DCPPA becomes eactly the algorithm proposed by [6]. If M = R n algorithm DCPPA reduces to the algorithm proposed in [30]. Therefore, the algorithm DCPPA on Hadamard manifolds is a natural generalization of the proimal point algorithm for DC functions on R n defined by Sun at al [30] and more general than proimal point algorithm proposed by Ferreira and Oliveira [6]. Now we shall establish the convergence of the algorithm. We begin by showing that algorithm DCPPA is a decent algorithm. Theorem 3 The sequence { k } generated by algorithm DCPPA satisfies: 1. either the algorithm stops at a critical point; 2. or f decreases strictly, i.e., f( k+1 ) < f( k ), k 0. Proof It follows from (8) and (9) that and w k = 1 ep 1 k y k h( k ) (10) 1 ep 1 k+1 y k g( k+1 ). (11) If k+1 = k the algorithm stops and, this clearly implies that 1 ep 1 y k k h( k ) g( k ), which means, k S. Now, suppose k+1 k. Using (10) and (11) in (7), we obtain that

7 A proimal point algorithm for DC functions on Hadamard manifolds 7 and h() h( k ) + 1 ep 1 k y k, ep 1, M k g() g( k+1 ) + 1 ep 1 k+1 y k, ep 1 k+1, M. Adding the last inequalities with = k+1 in the first one and = k in the second one, we have f( k ) f( k+1 ) + 1 [ ep 1 k ] y k, ep 1 k+1 + ep 1 k y k, ep 1 k+1 k. (12) k+1 Now, consider the geodesic triangle (y k, k, k+1 ) and set θ = (ep 1 k By Theorem 2, we have d 2 (y k, k ) + d 2 ( k, k+1 ) 2d(y k, k )d( k, k+1 ) cos θ d 2 (y k, k+1 ). Since ep 1 k y k, ep 1 k+1 = 2d(y k, k )d( k, k+1 ) cos θ, so that k ep 1 y k, ep 1 k k+1 1 k 2 d2 (y k, k ) d2 ( k, k+1 ) 1 2 d2 (y k, k+1 ). Similarly, considering the geodesic triangle (y k, k+1, k ) and setting θ = (ep 1 y k, ep 1 k ), we have k+1 k+1 y k, ep 1 k k+1 ). ep 1 k+1 y k, ep 1 k+1 k 1 2 d2 (y k, k+1 ) d2 ( k, k+1 ) 1 2 d2 (y k, k ). Adding the last two inequalities, we obtain that ep 1 k y k, ep 1 k+1 + ep 1 k y k, ep 1 k+1 k d 2 ( k, k+1 ). k+1 Combining the above inequality with (12), we have that this means that f( k+1 ) < f( k ). f( k ) f( k+1 ) + 1 d 2 ( k, k+1 ), (13) Corollary 1 Consider { k } generated by algorithm DCPPA, then the sequence {f( k )} is convergent. Proof Being f bounded from below due to the last theorem that the sequence {f( k )} is bounded and thus has at least one cluster point. Indeed, let {f( k )} admit two different cluster points f 1 < f 2. Let f( k j ) and f( k l ) be two subsequences converging to f 1 and f 2, respectively. Set ɛ = f 2 f 1 2, then there eist k j0, k l0 N such that f( k j ) < f 1 + ɛ f 2 ɛ < f( k l ), for all k j, k l k 0 = ma{k j0, k l0 }. By virtue of item 2 of last theorem, we have f( k j ) f( k 0 ) < f 1 + ɛ = f 2 ɛ. This is a contradiction, and hence {f( k )} has at the most one cluster point.

8 8 J.C.O. Souza, P.R. Oliveira Corollary 2 If f is a continuous function and { k } is bounded, then lim k f(k ) = f(), for some cluster point of { k }. Proof Let { k j } be any convergent subsequence with limit M. Since f is continuous, then f( k j ) f(). Thus, for a given ɛ > 0, there eists j 0 N such that j j 0, we have f( k j ) f() < ɛ. Since f is a decent function, we obtain that f( k ) f() = f( k ) f( k j 0 ) + f( k j0 ) f() f( k j0 ) f() < ɛ, k k j0, for an arbitrary ɛ > 0 and the proof is concluded. Proposition 5 Consider { k } generated by algorithm DCPPA, then. In particular lim k + d(k, k+1 ) = 0. Proof From (13), we have that and, therefore 1 d 2 ( k, k+1 ) f( k ) f( k+1 ) n 1 1 d 2 ( k, k+1 ) f( 0 ) f( n ). Since f is bounded from below and { } is bounded, we obtain, and it follows that lim k + d(k, k+1 ) = 0. d 2 ( k, k+1 ) < d 2 ( k, k+1 ) < According to Proposition 2, in Algorithm DCPPA, if { k } is bounded, then {w k } is also bounded. Since the eponential mapping is a diffeomorphism, we also have that {y k } is bounded. Theorem 4 Suppose that { k } is bounded. Then every cluster-point of { k } is a critical point of the function f. Proof Let and y be cluster points of { k } and {y k }, respectively. So, consider two subsequences k j and y k l converging respectively to and y, i.e., k j and y k l y. From definition of the sequences { k }, {y k } and { } [b, c], we have and h(z) h( k j ) + 1 c ep 1 k j yk l, ep 1 z, z M k j g(z) g( k j+1 ) + 1 c ep 1 k j +1 y k l, ep 1 k j +1 z z M. Taking k j, k l and making use of the Proposition 5, we have that 1 c ep 1 y h() and 1 c ep 1 y g(). In other words that is a critical point of f.

9 A proimal point algorithm for DC functions on Hadamard manifolds 9 Corollary 3 Suppose that { k } is bounded and S is a singleton, i.e., S = { }, then the entire sequence { k } converges to. Proof Suppose, by contradiction, that there eist an ɛ > 0, such that d( k, ) ɛ, (14) k k 0. Note that there eist a subsequence { k j } { k } such that k j. By the theorem we have S. But S = { }, and thus =. Therefore, for all ɛ > 0, there eist k 0 N such that d( k j, ) = d( k j, ) < ɛ, k k 0, violating (14). This completes the proof. Remark 1 If the level set of f and the subdifferential of h are compact and bounded, respectively, then the sequences { k } and {y k } are bounded. If f is strictly conve and coercive or strongly conve, then S is a singleton. Remark 2 It is worthwhile to point out that under the assumptions of Corollary 2, if f satisfies the sharp minima condition (see Polyak [35]), then the whole sequence { k } converges to some point S. Regarding weak sharp minima introduced by Ferris [36], it was recently considered on the contet of Riemannian manifolds by Li et. al. [16] and finite termination of Proimal Point Algorithm on Hadamard manifolds by Bento and Cruz Neto [37]. We hope that this paper may stimulate further research involving Algorithm DCPPA and these concepts. 5 Ineact Version Here we consider the approimate version obtained by replacing the eact subdifferential by the approimate one, since the function h and g are assumed to be conve, proper and lower semicontinuous. We define 0 h() = h() and 0 g() = g(), for any M. Furthermore, directly from the definition it follows that 0 ɛ 1 ɛ 2, then and ɛ1 h() ɛ2 h() ɛ1 g() ɛ2 g(). We recall that a vector w T M is called an ɛ-subgradient (with ɛ 0) of f at dom(f), denoted by w ɛ f(), if f(y) f() + w, ep 1 y ɛ, y M. Thus ɛ h() and ɛ g() are an enlargement of h() and g(), respectively. The use of elements in ɛ h() and ɛ g() instead of h() and g() allows an etra degree of freedom which is very useful in various applications. Setting ɛ = 0 one retrieves the eact subdifferential. To this reason we consider the following ineact version of Algorithm DCPPA: Algorithm (IDCPPA-1) Step 1: Given an initial point 0 M, a bounded sequence of positive numbers

10 10 J.C.O. Souza, P.R. Oliveira { } [b, c] and ɛ k 0. Step 2: Compute Step 3: Compute w k ɛk h( k ) and set y k := ep k( w k ). (15) k+1 : arg min M {g() d 2 (, y k )} 1 ep 1 k+1 y k ɛk g( k+1 ) (16) If k+1 = k, stop. Otherwise, k := k + 1 and return to Step 2. Theorem 5 Let { k } be a sequence generated by Algorithm IDCPPA-1. Suppose that { k } is bounded and + ɛ k <. Then the sequence {f( k )} is convergent and every cluster-point of { k } is critical point of the function f. Proof Similar to Theorem 3, we have that Then, f( k ) f( k+1 ) + 1 d 2 ( k, k+1 ) 2ɛ k. n 1 1 n 1 d 2 ( k, k+1 ) f( 0 ) f( n ) + 2 ɛ k. c Since f is bounded from below, the inequality above clearly implies that d 2 ( k, k+1 ) <, thanks to the summable assumption of {ɛ k }. Thus, lim k + d(k, k+1 ) = 0. Now, let and y be cluster points of { k } and {y k }, respectively. So, consider two subsequences k j and y k l converging respectively to and y, i.e., k j and y k l y. From definition of Algorithm IDCPPA-1, we have and h(z) h( k j ) + 1 c ep 1 k j yk l, ep 1 k j z ɛ k j, z M Since, that 1 c ep 1 g(z) g( k j+1 ) + 1 c ep 1 k j +1 y k l, ep 1 k j +1 z ɛ kl, z M. lim ɛ k = 0 and k lim k + d(k, k+1 ) = 0, taking k j, k l, we have y h() and 1 c ep 1 y g(). The proof is complete. As remarked in Rockafellar [3], for a proimal point method to be practical, it is also important that it should work with approimate solutions of the subproblems. To the best of our knowledge, approimate solutions of proimal points has not been found to be eplored in DC functions setting. For this reason, we provide an ineact proimal point algorithm for DC functions with a relative error tolerance. Algorithm (IDCPPA-2) Step 1: Given an initial point 0 M and a bounded sequence of positive numbers

11 A proimal point algorithm for DC functions on Hadamard manifolds 11 { } [b, c]. Step 2: Compute Step 3: Compute where w k h( k ) and set y k := ep k( w k ). (17) e k+1 g( k+1 ) 1 ep 1 k+1 y k (18) e k+1 ηd( k+1, k ), η [0, 1). (19) If k+1 = k, stop. Otherwise, k := k + 1 and return to Step 2. When k+1 = k or η = 0, (19) obviously implies that e k+1 = 0 and the Algorithm IDCPPA-2 reduces to the Algorithm DCPPA. Theorem 6 Let { k } be a sequence generated by Algorithm IDCPPA-2. Suppose that { k } is bounded, then the sequence {f( k )} is convergent and every clusterpoint of { k } is a critical point of the function f. Proof Similar to Theorem 3, we have which by (19) imply that f( k ) f( k+1 ) + 1 d 2 ( k, k+1 ) + e k+1, ep 1 k+1 k, f( k ) f( k+1 ) + (1 η) d 2 ( k, k+1 ) > f( k+1 ), if k+1 k. Otherwise, the algorithm stops. Since f is bounded from below, we have that the sequence {f( k )} is convergent. Furthermore, (1 ηc) c n 1 The inequality above obviously implies that d 2 ( k, k+1 ) f( 0 ) f( n ). d 2 ( k, k+1 ) <. Thus lim k + d(k, k+1 ) = 0. Now, let and y be cluster points of { k } and {y k }, respectively. So, consider two subsequences k j and y k j converging respectively to and y (here we will use the same notation for the inde even if it needs etracting other subsequences), i.e., k j and y k j y. From definition of the Algorithm IDCPPA-2, we have and h(z) h( k j ) + 1 ep 1 c k j yk j, ep 1 z, z M k j kj g(z) g( k j+1 ) + 1 j ep 1 k j +1 y k j, ep 1 k j +1 z + e k j+1, ep 1 k j +1 k j, z M.

12 12 J.C.O. Souza, P.R. Oliveira By passing to the limit in the above relations, since lim k ek = 0, lim k + d(k, k+1 ) = 0 and taking into account the fact that the functions g, h are lsc and { } is bounded, we have that h() g(), in other words that is a critical point of f. 6 Eample and application In this section we present an eample of a nonconve minimization problem where the objective function is defined on the Poincaré half plane (a Hadamard manifold with curvature identically to 1). In the eample, the proimal point algorithm proposed by Ferreira and Oliveira [6] does not apply. However, the method proposed in this article applies. Also, an application to constrained maimization problems on Hadamard manifolds is given. 6.1 Eample Consider the Poincaré upper half-plane H = {(u, v) R 2 : v > 0} endowed with the Riemannian metric defined for every (u, v) H by g ij (u, v) = 1 v 2 δ ij, for i, j = 1, 2. The pair (H, g) is a Hadamard manifold with constant sectional curvature 1 and the geodesics in H are the semi-lines and the semicircles orthogonal to the line v = 0 (see [34] page 20) with the following natural parameterizations γ a : u = a, v = e s, s (, + ); γ b,r : u = b rtanh s, v = r, s (, + ). cosh s The geodesic passing at moment s = s 0 through the point p = (, y) tangent to the vector w = (u, v) T p H is { (, ye s s 0 ), for ) u = 0, v = y; γ(s) = vu w 2 ( + w + u tanh(s), y w 1 u, for u 0, cosh(s) where s [s 0, ) and w = u 2 + v 2. Consider the geodesic passing at moment t = 0 through the point p = (, y) tangent to the vector w = (u, v) T p H. Hence, the eponential map is defined by { ep p w = γ(1) = ( + vu w + w 2 u tanh(1 + s 0), y w u (, ye), for ) u = 0, v = y;, for u 0. 1 cosh(1+s 0 ) The Riemannian distance between two points (u 1, v 1 ), (u 2, v 2 ) H is given by d((u 1, v 1 ), (u 2, v 2 )) = arccosh (1 + (u 2 u 1 ) 2 + (v 2 v 1 ) 2 ). 2v 1 v 2 Let f : H R be a function given by f(, y) = 4 + y y Note that f is bounded from below and f is not a conve function on H while g(, y) = 4 + y 4 and h(, y) = y are conve functions on H. Clearly, the set of critical points of f is nonempty. Therefore, f agree with the assumptions made. Thus, the Algorithm DCPPA can be applied.

13 A proimal point algorithm for DC functions on Hadamard manifolds Application to constrained maimization problems We consider the problem of maimizing a conve lower semi-continuous function h on a closed conve set C M, namely ma h(). (20) C This problem can be rewritten as a DC problem, and (20) is equivalent to the following problem: min M {δ C() h()}, (21) where δ C () is the indicate function defined by δ C () = 0 if C and δ C () = + otherwise. Let N C () denote the normal cone of the set C at a point C: Then N C () := {u T M; u, ep 1 y 0 y C}. δ C () = N C (), C. In this contet Algorithm DCPPA takes the following form: Compute w k h( k ) and set y k = ep k( w k ). Define k+1 M as the solution of the following variational inequality problem: ep 1 k+1 y k, ep 1 k+1 y 0, y C. Eistence and uniqueness theorems for variational inequalities on Hadamard manifolds can be found, for instance in [11], [13]. Acknowledgements The authors wish to epress their gratitude to the anonymous referee for his helpful comments. References 1. Martinet, B., Regularisation d inéquations variationelles par approimations succesives, Rev. Franaise d Inform. Recherche Oper., 4, (1970) 2. Moreau, J.J., Proimité et dualité dans un espace Hilbertien, Bull. Soc. Math. France, 93, (1965) 3. Rockafellar, R.T., Monotone operators and the proimal point algorithm, SIAM J. control. optim., 14, (1976) 4. Bento, G.C., Ferreira, O.P., Oliveira, P.R., Local convergence of the proimal point method for a special class of nonconve functions on Hadamard manifolds. Nonlinear Anal. 73, (2010) 5. Papa Quiroz, E.A.., Oliveira, P.R., Proimal point method for minimization quasiconve locally Lipschitz functions on Hadamard manifolds. Nonlinear Anal. 75, (2012) 6. Ferreira, O.P., Oliveira, P.R., Proimal point algorithm on Riemannian manifolds, Optimization, (2002) 7. da Cruz Neto, J.X., Ferreira, O.P., Lucambio Pérez, L.R., Németh, S.Z.: Conve-and monotone- transformable mathematical programming problems and a proimal-like point algorithm. J. Glob. Optim. 35, (2006) 8. Ferreira, O.P., Oliveira, P.R.: Subgradient algorithm on Riemannian manifolds. J. Optim. Theory Appl. 97, (1998) 9. da Cruz Neto, J.X., de Lima, L.L., Oliveira, P.R.: Geodesic algorithms in Riemannian geometry. Balk. J. Geom. Appl. 3, (1998)

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